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The time required to determine the larger of 2 digits decreases with their numerical distance, and, for a given distance, increases with their magnitude (Moyer & Landauer, 1967). One detailed quantitative framework to account for these effects is provided by random walk models. These chronometric models describe how number-related noisy partial evidence is accumulated over time; they assume that the drift rate of this stochastic process varies lawfully with the numerical magnitude of the digits presented. In a complete paired number comparison design we obtained saccadic choice responses of 43 participants, and analyzed mean saccadic latency, error rate, and the standard deviation of saccadic latency for each of the 72 digit pairs; we also obtained mean error latency for each numerical distance. Using only a small set of meaningfully interpretable parameters, we describe a variant of random walk models that accounts in considerable quantitative detail for many facets of our data, including previously untested aspects of latency standard deviation and error latencies. However, different from standard assumptions often made in random walk models, this account required that the distributions of step sizes of the induced random walks are asymmetric. We discuss how our findings can help in interpreting complex findings (e.g., conflicting speed vs. accuracy trends) in applied studies which use number comparison as a well-established diagnostic tool. Finally, we also describe a novel effect in number comparison, the decrease of saccadic response amplitude with numerical distance, and suggest an interpretation using the conceptual framework of random walk models.
We present a new quantitative process model (GSDT) of visual search that seeks to integrate various processing mechanisms suggested by previous studies within a single, coherent conceptual frame. It incorporates and combines 4 distinct model components: guidance (G), a serial (S) item inspection process, diffusion (D) modeling of individual item inspections, and a strategic termination (T) rule. For this model, we derive explicit closed-form results for response probability and mean search time (reaction time [RT]) as a function of display size and target presence/absence. The fit of the model is compared in detail to data from 4 visual search experiments in which the effects of target/distractor discriminability and of target prevalence on performance (present/absent display size functions for mean RT and error rate) are studied. We describe how GSDT accounts for various detailed features of our results such as the probabilities of hits and correct rejections and their mean RTs; we also apply the model to explain further aspects of the data, such as RT variance and mean miss RT.